Edge-magic labellings
Atilio Luiz
MO804 - Topicos em Teoria de GrafosUNICAMP, Brasil
September 28, 2017
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Magic Square
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◮ A magic square is a n× n square grid filled with distinct positiveintegers in the range 1, . . . , n2 such that each cell of the gridcontains a different integer and the sum of the integers in each row,column and diagonal is equal.
Figure: (Lo Shu square) The smallest (and unique up to rotation andreflection) magic square of dimension 3× 3.
Magic Square in Art
Figure: (Melencolia I, Albert Durer, 1514) The sum 34 can be found inthe rows, columns, diagonals, each of the quadrants, the center foursquares, and the corner squares (of the 4× 4 as well as the fourcontained 3× 3 grids).
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Srinivasa Ramanujan’s magic square
The Indian mathematician Srinivasa Ramanujan created a square wherethe first row shows his date of birth, 22nd Dec. 1887.
Figure: This magic square has 24 groups of four fields with the sum of139 and in the first row - shown at bottom-right - Ramanujan’s date ofbirth.
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Origins of magic graphs
A.Kotzig
G. Ringel
J. Sedlacek
Smolenice 1963
Figure: Smolenice Symposium in Graph Theory, Czechoslovakia, 1963.
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Two famous problems of the Smolenice Symposium
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Problem 27
◮ [Jiri Sedlacek, 1963] A simple connected graph G is called magic ifthere is a real-valued labelling of the edges of G such that:
(i) distinct edges have distinct values;(ii) the sum of values assigned to all edges incident to a given
vertex v ∈ V (G) is the same for all vertices of G.
◮ Find necessary and sufficient conditions for a graph to be magic.
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Edge-magic labellings
◮ [Kotzig and Rosa, 1970] Let G be an (n,m)-graph. An edge-magiclabelling of G is a bijection f : V (G)∪E(G) → [1,m+ n] such thatf(u) + f(uv) + f(v) = k, for all uv ∈ E(G).
◮ If G has an edge-magic labelling, then G is called edge-magic.
◮ The constant k is called the valence of the labelling f .
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Complementary labelling
◮ The complementary labelling of an edge-magic labelling f , denotedf , is defined by f(x) = n+m+1− f(x), for all x ∈ V (G)∪E(G).
◮ f is also an edge-magic labelling and its valence isk = 3(n+m+ 1)− k.
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Warm up
Theorem
Let G be a (n,m)-graph. If G has m even and n+m ≡ 2 (mod 4), andevery vertex of G has odd degree, then G has no edge-magic labelling.
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Warm up
Theorem
Let G be a (n,m)-graph. If G has m even and n+m ≡ 2 (mod 4), andevery vertex of G has odd degree, then G has no edge-magic labelling.
Corollary
The complete graph Kn is not edge-magic when n ≡ 4 (mod 8). Thewheel Wn with n+ 1 vertices is not edge-magic when n ≡ 3 (mod 4).
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Stars
Theorem
Every star is edge-magic.
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Stars
Theorem
Every star is edge-magic.
Corollary
In any edge-magic labelling of a star K1,n, the center receives label 1,n+ 1, or 2n+ 1.
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Stars
Theorem
Every star is edge-magic.
Corollary
In any edge-magic labelling of a star K1,n, the center receives label 1,n+ 1, or 2n+ 1.
Theorem
There are 3 · 2n edge-magic labellings of K1,n up to equivalence.
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Trees
Conjecture [Kotzig and Rosa, 1970]
Every tree is edge-magic.
Theorem [Kotzig and Rosa, 1970]
Every caterpillar is edge-magic.
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Linear forests
Theorem
The one-factor F2n is edge-magic if and only if n is odd.
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Cycles
Theorem
All cycles Cn with n ≥ 3 are edge-magic.
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References I
[1] J. A. Gallian.A dynamic survey of graph labeling.The Electronic Journal of Combinatorics, 2016.
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